NONLINEAR GRAVITY WAVE-WIND INTERACTIONS
AND JET STREAM GRAVITY WAVE GENERATION
by
DONALD LEE PAUL
B.S., Massachusetts Institute of Technology
M.S., Massachusetts Institute of Technology
SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
January 1977
Signature of Author
Department of Earth & Planetary Sciences
Certified by
-Thesis Supervisor
Accepted by
Chairman,
Departmental Committee on Graduate Students
MIT Ll RAR
77
NONLINEAR GRAVITY WAVE-WIND INTERACTIONS AND
JET STREAM GRAVITY WAVE GENERATION
by
Donald Lee Paul
Submitted to the Department of Earth and Planetary Sciences
in January 1977, in partial fulfillment of the requirements
for the degree of Doctor of Philosophy.
ABSTRACT
Theoretical gravity wave source models have been
developed to explain observed ground pressure fluctuation
measurements associated with overhead jet streams.
The first
theory was developed using a multiple-scale analysis for
finite amplitude gravity waves in a sheared flow.
This anal-
ysis is equivalent to a nonlinear WKB approximation and calculations show the assumption of spatial smoothness of the
medium will be violated for the low Richardson numbers
associated with jet stream source regions.
A quasi-linear
theory was next developed which is not limited in applicability to large Richardson number shears.
The quasi-linear
theory allows the essence of the nonlinear wave-wind interaction to be explored, yet retains the features of linear
wave propagation away from critical levels.
For transient
gravity wave sources in the jet stream, numerical experiments
show two major effects.
First, the initial disturbance is
amplified in the source region at the expense of the wind and
gravity waves are radiated from the source region up the jet
and down toward the ground.
An observed peak to peak surface
pressure fluctuation of 0.5 millibar would correspond to maximum peak to peak sources region wave wind velocities of
approximately 6 m/s.
The second effect is a transfer of wind
energy from the upper critical level to the lower critical
level.
This results from the upper critical level absorbing
negative momentum waves that propagate up from the source
region, while the lower critical absorbs positive momentum
waves that propagate down to and are reflected back from the
ground.
The total energy transferred can be considerably
larger than the original disturbance energy when the source
region Richardson number is low.
A possible mechanism for generating the initial disturbances involves the generation of propagating subharmonics
by their nonlinear wave-wave interaction with an unstable
mode associated with shears having Richardson numbers less
than 0.25.
This possibility was studied using linear stabil-
ity theory for stratified shear flows and an approximate
theory for nonlinear wave-wave interactions.
The time his-
tory predicted for this interaction is compatible with that
of the subharmonic-mean wind interaction previously modeled.
The more complete modeling involving the interactions of the
unstable mode, the subharmonic, and the mean wind remains as
an extension of the current study.
iii
The unstable modes have
periods less than the Brunt period and the periods of their
subharmonics will be from 1.2 to 1.6 Bri*nt periods which
falls within the range of commonly observed jet stream
gravity wave spectra.
iv
TABLE OF CONTENTS
Page
ABSTRACT. .
.
. .
.
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.
. .
..
.
. .
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.
. .
. .
.
.
ii
.
.
vii
. .
1
.
.
6
Mathematical Development for Finite Amplitude
Waves . . . . . . . . . . . . . . . . . . . . . .
6
LIST OF FIGURES . . .
INTRODUCTION
I.
. . .
. .
. .
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.
.
. .
.
. .
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. .
. .
. .
. .
. . .
. .
. .
.
. .
. .
MULTIPLE-SCALE ANALYSIS OF NONLINEAR GRAVITY WAVES
.
IN A BOUSSINESQ FLUID WITH A SHEARED MEAN FLOW
A.
B.
C.
II.
III.
Application of the Method of Characteristics
to Low Frequency Gravity Waves. . . . . . . . .
.
17
Limitations of Multiple-Scale Analysis and the
Question of Partial Reflections . . . . . . . . .
20
HORIZONTAL FOURIER EXPANSION ANALYSIS:
A QUASI-LINEAR THEORY . . . . . . . . . . . .
A.
Development of the Governing Equations
B.
Stream Function - Vorticity Formulation .
. .
.
.
24
.24
. .
NUMERICAL EXPERIMENTS: WAVE-WIND INTERACTIONS
FOR TRANSIENT GRAVITY WAVE SOURCES
. . . . . .
. .
. .
.
.
30
.
. .
33
.
. .
33
A.
General Computational Procedure . .
B.
Basic Jet Stream and Gravity Wave Source
Models
. . . . . . . . . . . . . . . . . . . . .
39
Transient Behavior of Finite Amplitude
Gravity Wave Sources in Low Richardson
Number Shears . . . . . . . . . . . . . .
C.
IV.
.
.
. .
.
44
THE GENERATION OF JET STREAM GRAVITY WAVE SOURCES
BY WAVE-WAVE INTERACTION . . . . . . . . . . . . .
.
59
.
59
. .
61
. .
. .
.
. .
.
. . .
A.
Introductory Remarks
B.
Brief Review of Linear Stability Theory in
Stratified Shear Flow . . . . . . . . . . . .
C.
Source Generation by Subharmonic Wave-Wave
Interaction . . . . . . . . . . . . . . . .
. .
. .
.
63
TABLE OF CONTENTS (continued)
Page
V.
SUMMARY . .
. .
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.
.
76
APPENDICES
A.
B.
C.
Numerical Determination of the Eigenvalues
and Eigenfunctions of the Taylor-Goldstein
Equation . . . . . . . . . . . . . . . .
Brunt Period Reflection Conditions for
Unstable Stratified Shear Flows .
.
82
.
.
.
.
Detailed Source Region Waveforms
for Model A . .
.
..........
.
. .
89
91
. .
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.
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. .
.
105
. . .
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.
108
ACKNOWLEDGEMENT . . .
.
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.
. .
.
REFERENCES
BIBLIOGRAPHY
.
BIOGRAPHICAL NOTE .
vi
. .
. .
.
.
. .
. .
l.11
., 112
LIST OF FIGURES
Figure
Page
1.
Change in Mean Wind in M/S
. . . .
.
.
.
22
2.
S/F, the Absolute Value of the Ratio of the
Second Harmonic in u, as a Function About
100m above zc (from Breeding (1971)
.
.
.
28
Schematic of Model Jet Stream Profile
Including Regional and Source Subregional
Configuration . . . . . . . . . . . . . . .
.
.
.
39
.
.
.
47
.
.
.
48
.
.
.
49
3.
. .
. .
.
. .
.
4.
Time Energy Curves for Model A
5.
Time Net Energy Curves for Model A
6.
Time Energy Curves - Richardson Number
Effects . . . . . . . . . . . . . . . .
. .
.
. .
7.
Time Energy Curves - Source Width Effects
.
.
.
50
8,
Source Region Wave Energy: Amplification
Factor vs. Mean Initial Energy Density .-.
.
.
.
55
Net Energy Transport by Gravity Waves as a
Function of Source Subregion Richardson
Number
. .
. .
.
.
.
56
.
.
.
57
9.
10.
Maximum RMS Source Region Wave Velocity
in M/S . . . . . . . . . . . . . . . . .
.
11.
Smoothed Typical Gravity Wave Spectra (Normalized)
Observed at WEston, Mass. and Theoretical Values
From Subharmonic Generation . . . . . . . . . . . 60
12.
Stability Diagram for Hyperbolic Tangent Profile
and Computed Eigenvalues
. . . . . . . . . . . .
65
13.
Normalized Growth Rate vs. Wave Number for Unstable
66
Modes of the Hyperbolic Tangent Profile .....
14.
Time Behavior of Primary and Subharmonic
. . . . . . . . . . . . . . . . . . . .
Energies
72
Subharmonic Energy Spectrum - Maximum
Output Values . . . . . . . . . . . . . .
73
15.
vii
. .
. .
NONLINEAR GRAVITY WAVE-WIND INTERACTIONS AND
JET STREAM GRAVITY WAVE GENERATION
INTRODUCTION
Ground level pressure fluctuation measurements with an
array of micro-barographs have shown that most of the fluctuation energy in the period range of minutes to an hour behave
as propagating atmospheric gravity waves.
These waves are
especially strong when a well developed tropospheric jet
stream is located overhead.
Linear wave theory is able to
account for many features of these waves.
The frequency
spectrum of the waves shows a sharp high frequency cutoff at
the Brunt frequency.
This was shown to be due to the evanes-
cent wave behavior for waves with frequencies above the Brint
frequency and which have subsonic horizontal phase velocities.
The phase velocity of the waves can be used to determine the
height of the source region, since one can assume the source
region is convected with the mean wind, and in the troposphere
the wind velocity is usually monotonically increasing with
height.
On examining the upper air wind and temperature
records it was found that the source regions represented
those levels of the jet that were least stable as based on
the Richardson number (Claerbout, 1967; Madden and Claerbout,
1968).
Linear theory is unable to predict the source charac-
teristics, however, as the theory breaks down in the vicinity
2
of the critical level, that level where the wind velocity
equals the wave phase velocity.
Linear theory predicts
growing wave amplitudes and decreasing vertical wavenumbers
as one approaches tihe critical level (Booker and Bretherton,
1967; Claerbout, 1967), so that unstable wind shears would
result from the waves themselves.
Linear theory also predicts
that the wave is able to interact with the wind at the critical level (Bretherton, 1968; Garret, 1968; Hazel, 1967), and
this interaction could greatly alter the resulting wave
behavior.
Since linear theory cannot predict how the inter-
action proceeds, one must incorporate nonlinear terms in
order to study the real situation existing in the-jet stream
source regions.
A nonlinear investigation of gravity wave behavior
near the source region and the relation to ground pressure
observations and energy transport within the jet stream
seemed a worthy topic of investigation since waves involving
surface pressure variations of only fractions of a millibar
could become important in the jet stream due to the strong
and nonlinear behavior near critical levels.
In addition,
the association of the source regions with low Richardson
numbers was an indication that a study of possible gravity
wave source generation by shear flow instability would also
be valuable.
A start on such a nonlinear study of gravity wave-wind
interactions was made by Breeding (1971).
In that study the
nonlinear critical level phenomenon was investigated by means
of finite difference equations.
energy and momentum is
The study showed how wave
taken up by the mean wind and how some
higher harmonic wave generation also results.
Unfortunately,
the finite difference calculations require large amounts of
computer time, and without a more analytic basis it is difficult to generate a complete understanding of the phenomenon.
In addition, there remains the question of source generation
and transient behavior in full jet stream models.
Breeding's
calculations did show, however, that with realistic wave
amplitudes, the time scale of the changes in the waves and
the wind could be slow relative to a wave period.
This re-
sult could justify certain simplifications in the analysis
and opens up the possibility of using more analytic methods
to efficiently study these interactions.
In Section I a multiple scale analysis is applied to
gravity wave propagation in a sheared flow.
This analysis is
based on the concept that while the wave amplitudes are
finite so that second-order products cannot be neglected, the
changes in the system are slow relative to the wave and
spatially smooth compared to the wave phase structure.
After
considerable algebraic manipulation one is led to a much
simpler set of equations for the waves and mean flow that is
equivalent to a nonlinear WKB approximation for gravity wave
propagation.
Unfortunately, in the jet stream environment
the condition of spatial smoothness is not upheld, and this
analysis gives unrealistic results.
The important parameter
to define spatial variations is the wave impedance change
per radian.
From linear theory we know that this parameter
is inversely proportional to the square root of the
Richardson number, and is therefore not small in regions of
interest where the waves are generated.
Breeding's analysis also showed that the wave harmonic
generation is much smaller than the mean flow - fundamental
interaction, and thus an expansion in horizontal Fourier components is justified.
II.
Such an approach is taken up in Section
This method also leads to simplified numerical calcula-
tions, but in this case good agreement is obtained with
Breeding's two-dimentional finite difference calculations.
A
similar expansion was also developed by Boer (1970).
This quasi-linear analysis can be given a rather
simple interpretation.
On
the whole the waves behave as in
linear theory, but in a time-varying medium.
The spatial
and temporal variations of the medium are straightforwardly
related to nonlinear wave terms.
From linear theory we are
aware of the large reflection effects occurring because of
rapid impedance changes with height in regions of low
Richardson number, and it is the presence of these partial
reflections which invalidated the multiple-scale analysis.
These reflections, which are included in the quasi-linear
theory, greatly influence the resulting wave-wind interactions.
In Section III the quasi-linear theory is used in a
numerical study of the behavior of finite amplitude gravity
wave transients in a jet stream model as a function of source
5
characteristics and source region Richardson number.
Calcu-
lations will show that gravity waves provide a very efficient
coupling between the source region critical level and the
critical level at the top of the jet.
The steady state
result is a transfer of wind energy from the topside to the
underside critical level, the amount of energy transport
rapidly increasing for decreasing Richardson number.
Calcu-
lations will also show that source region wave-wind velocities of a 6 m/s can produce the observed pressure fluctuation
amplitudes of 0.5 millibar at the ground.
In the final section we examine the question of
gravity wave source generation itself.
From linear stability
theory for stratified shear flows we know that a certain
range of wavenumbers will be unstable and grow at the expense
of the wind if the Richardson number is less than 0.25
(Collyer, 1970; Drazin, 1958; Drazin and Howard, 1966; Hazel,
1972; Howard, 1963; Jones, 1968; Miles, 1961 and 1963;
Richardson, 1920).
However, these unstable wave modes have
frequencies above the BrUnt frequency cutoff and are trapped
in the jet stream.
Untrapped waves can be generated through
wave-wave interaction among unstable modes.
The development
of such waves is studied in Section IV using linear stability
theory and an approximate nonlinear wave-wave interaction
formulation.
The analysis shows that such effects are a
possible explanation for the generation of observed gravity
waves.
I. MULTIPLE-SCALE ANALYSIS OF NONLINEAR GRAVITY WAVES IN A
BOUSSINESQ FLUID WITH A SHEARED MEAN FLOW
A.
Mathematical Development for Finite Amplitude Waves
We wish to examine in detail one method of approxi-
mation to deal with finite amplitude gravity waves interacting with a sheared mean flow.
In an attempt to eliminate
algebraic complexities without altering the basic nature of
the problem, we shall assume the equations of motion are
those describing a Boussinesq fluid.
The use of the Bous-
sinesq approximation implies that the speed of sound and
scale height of the mean density are very large compared to
the phase velocities and vertical wavelengths of the fluid
motions (Spiegel and Veronis, 1960).
The mean vertical den-
sity structure is then described by the Brunt-Vasailia
frequency QAB, and is otherwise considered a constant.
We then wish to examine the interaction of a train of
internal gravity waves with a mean horizontal flow.
The mean
flow is assumed to be sheared and initially independent of
time; however, nonlinear interactions with the wave train
will introduce both time and spatial variations on the initial
structure.
The governing equations and variables are given as
follows:
(1.1)
.
+
H~~)l
~L+?
±
0
+U
(1. 4)+DI4
0
where
u
= Horizontal wave velocity field
U
= mean horizontal velocity field
w
= vertical wave velocity field
P
= (wave pressure field) /p
p
= mean density field
$
= wave bouyancy function
~B2
CI
For future reference, it can easily be shown that an
energy conservation equation can be derived from (1.1)
-
(1.4)
as:
(1.
5) ~..~Fu.RV
where
E =
i~
+ WZ +(i.+-O
+
/IN.
The nonlinear terms in the Reynold's stress in Equations (1.1) - (1.4) make a general solution to the above system virtually impossible,
However, under certain approxima-
tions one can look for special solutions which may give
insight into the particular problem.
is that of multiple-scale analysis.
One method available
This method is essen-
tially a generalized nonlinear WKB approximation for application to hyperbolic partial differential equations, and has
been shown (Whitham, 1970)
to be equivalent to the
The
phase-averaged techniques developed by Whitham (1965).
method of multiple scales has been applied to general nonlinear one-dimensional dispersive wave systems by Nayfeh and
Hassen (1971), and by Grimshaw (1972) to gravity waves in
fluids without sheared mean flows.
The following develop-
ment represents a new application of the method to the study
of the propagation finite amplitude internal gravity waves
and their critical level interaction with a sheared mean flow.
The basic assumption of the multiple-scale analysis
is that there exist two space-time regimes, one representing
the rapid phase fluctuations in the wave train, and one
representing variation of such variables as amplitude,
frequency, wavenumber, and properties of the medium.
In
order to describe mathematically the multiple-scale procedure,
we introduce a small parameter e << 1, such that
(1.6)
X
&.y. , T * Ct
where (x,t) refer to the short-term phase behavior of the
wavetrain and (X,T) to the long-term behavior of the wave
amplitude, phase derivatives (frequency and wavenumber) and
medium properties.
In linear theory one usually deals with
the derivatives of the phase 6 = wt - k'x.:
In the multiple-
scale analysis a parallel development follows as the variables (x,t) are introduced through a rapidly-varying phase
variable 6.
written as:
The time and spatial derivatives can then be
(1.7)
a)
O -
'
+ E.
=,
b)
+
c)
..
as
-az
In general, the frequency o and wavenumbers k and k
Finally, we
are also functions of the variables (X, Z, T).
introduce an asymptotic series in powers of g. for each
The
dependent wave variable as functions of (e,X,T).
specific notation will be as follows:
(1.8)
a)
b)
c)
'.(Q,
,
T)
+ E
--
()
W = w ( 2ST) -t-g,W1 ( 1
I> = PO (9 XT) +
.P (&,LT)+
.T)+
A
d)
IT) +-
)
It should be noted that the wave field variables are
not considered small, that is of O(c), so that quadratic
products of zero order in the governing equations may not be
neglected as is the case in linear theory.
Furthermore,
since the wave-train amplitude is not to be considered 0(c)
relative to ambient values, one would also expect wave-wind
and wave-wave interactions to produce changes in the medium
properties.
The total field variables will then be written
as:
(1.9)
a)
b)
t
14,
W
U(Z) +
MW(X)T) +
6 W(K T) +WO(G
\o((,XT)
+ O(E)
IT) +0 QCf.)
c)
P = AP (.,T) + 1 (el;T) + o
d)
4P
= A4 (KbX T) +
c1
0(&,j&L 1-)+
0 oC~
Note that we initially assume only a horizontal
shear U(z) and hydrostatic equilibrium so that P (z) and
T(z) do not appear in the Boussinesq equations (1.1)and in (1. 9) c,
(1.4)
d.
Zero and First Order Equations
Upon substitution of the scaled derivatives and the
asymptotic series for the field variables into the governing
equations, one obtains a sequence of systems of partial
differential equations after ordering by powers of
e.
Of
immediate interest are the systems of equations corresponding
The Q(e") system contains only deriva-
to 0(co) and 0(W).
tives with respect to 6 and gives the phase dependence of the
leading terms of the wave field variables.
The O(e)system
is as follows:
uo+U+
t.)(w+
(1.10)
'(w A
(1.11)
-.
-R(w+sw
(1.12)
Aw)=
(). +1 -A) (Vo + AW) --- (Pot+AP)
+
4>+
s
[+
[&W'(+ 0
+e
-Q
7->)
O+A
In the above equations o,K,2, have been collected
inside the brackets as they are a function only of (X,T) and
not 6. Also, since U, Au, Aw, AP and A$ are not functions of
6, some terms could be deleted but are retained for convenience in the manipulations that follow.
Equations (1.10) and (1.13) can be integrated with
respect to 6 directly.
(1.14)
fir. ((AO
From (1.13),
U+(A)
-
.(WO+A))
-
and from (1.10)
(1. 15)
(l4
(0-'-
+ A(
Eliminating (Pet
(
) and (0+&-)
o A
-
'-
k o(._.T).
in favor of (WD t.AW ) in
(1.11) gives
(1.16) (-
(I &A/&+AtA/)
and finally eliminating
+
J'+
(4t &4 ) yields
If we normalize the period of 6 to
(1.18)
A>tW
&J
=
\
T f
\
2Tr,
the solution is
T) e>
which requires from ( 1.17) that
(1.19)
tC..QA
The relation (1.19) defines a "local" dispersion relation and
if f0 (X,T) = KU(z), one would have the linear dispersion
relation for internal gravity waves in a Boussinesq fluid
with horizontal shear flow.
Since w0 = w (6,X,T) and
Aw = Aw(X,T), we can take Aw = 0 and then
(1. 20)
WOG
(G)I--X-r
~o(~)=.W~ri
+
WCi T)e
Referring back to Equation (1.14), we can write
(1.21)
+
[XITA
and since u0 = u0 (O,X,T) this can only hold for all 6,X,T if
(1.22)
u (6,X,T) =
(Wei+ W e
)
and
(1.23)
f0 (XT)= K(U(z) +Au(X,T)].
We observe then that the "local" dispersion relation
(1.15) is exactly analogous to the dispersion relation of
linear theory wherein the initial mean horizontal flow U(z)
is replaced by the instantaneous mean horizontal flow
U(Z) + Au(X,T)..
The mean flow is modified by the term Atk(X,T)
resulting from nonlinear wave-wave interaction, and this will
in turn modify the wave train via wave-wind interaction.
The remaining amplitude relationships can be determined straight away.
(1. 24)
(1.26)
Ne +We
+~a-f(
IC PO
-
which in turn allows us to take A$ = 0.
The amplitude and
polarization relations are exactly analogous to those for
linear theory in terms of the lowest order wavefield variables.
The effect of the finite amplitude of the wave train is
13
observed in the contribution to altering the mean horizontal
flow and mean pressure field, which in turn will modify the
wave train itself.
The 0(cS) system has yielded the phase
relationships within the wavetrain and the 0(0') system will
give the dependence of the zero-order solution on (X,T).
The 0(e') system is:
(1.27)
+~
+Uc +tt
+
~+
L)WAcI
A I IT)
(CG(G
(1.
29)
-[
(?-.-
(1.30
+- [c1
1
l(,
We observe that the homogeneous part of the system of
partial differential equations for u1 , w1 , Pl and $l is identical to that of the corresponding 0 (g") variables .
In
general then, we can expect terms which are secular (nonperiodic) in 6 to be generated in the inhomogeneous solution
to the system (1.27) - (1.30).
To maintain an asymptotic
series which will be uniform in 0, at least to 0(6,1), certain
necessary conditions must exist concerning the inhomogeneous
portion of the 0()
equations .
The removal of secular terms
at 6(e.) will generate the equations required to describe the
m)
behavior of the 0(eo) variables with respect to (XT).
Since there are four independent equations, in general
we would expect four conditions and this proves to be the
case.
Immediately from equations ( 1.27) and (1.30), we see
that removal of secularities requires
q 9
(1. 31)
=' o
T
0.
and
(1.32)
SdG b(Q T)
=0O
0
Explicitly this requires
[WW
(1.33)
+APl +
+2
L2
We]*
and
(1.34
X
To find the remaining two secular conditions, we
derive the governing second-order differential equation for
W1 :
-
~6E
1~
Ai
A,
.o E (,.T)
Since
+
we are reduced to the form:
=1from the 0'(e) calculations,
W
(1.36)
-E
The two remaining secular conditions follow directly.
Multiplying both sides of W and integrating over
e from
0 to 27T gives
and since W9 and - -4 are periodic over 2rr and Wand1
are also
(1. required
periodic over 27r (with secularities
38WeW
,(,to be
LT)=)o
removed) then
3wo
awl
21L
O
In an exactly analogous way, multiplying by - integrating (1.36) over 2w gives:
dG W* E,(G,
(1.39)
-)
and
i
0.
As a simple example, consider the case where
E1 (e,X,T)
=
a(_X,T)W9 + 6(X,T) ----
conditions (1.38) and (1.39) give
s= a
solution to (1.36) using E
s= W (1+S/2) +
secular terms in 0 unless a
The secular term removed
.
0.
=
The exact
as above is
-l (-aco/2)
=
=
which clearly will have
0 is required.
The derivation of the amplitude equations for the
zero-order field variables based on removal of secular terms
in 0 provides the mathematical foundation for a more intuitive approach.
Whitham (1965) has shown that if the original
equations can be put into conservation form, the amplitude
equations can be derived by simply phase-averaging.
A con-
servation equation will have the form
(1.
4
BM
0) -+
-Fx
ZF
-+...
a T~
Zx
and for the system (1.1) - (1.4),
,
az
there are three conservation forms.
The governing amplitude
equations are:
-+
(1.41)
(< e
--
>
3x
lA/0
-
(horizontal momentum conserva-
tion)
< EQ%
(1.42)
+ <(oEc7
+ (± +AU
E<-
where
Po
(oo
0
--
+
(energy conservation)
[W. Ec). + <
+
(1.43)
(Eo)e
0
4 (A4
-(.
(mass conservation)
'eb x
In addition there are two phase conservation
equations,
- -
(1.44)
(1.45)
3T
.
2.
and one dispersion relation
(1.46)
-.
+
.=
which complete the system.
A particularly important -case arises when the amplitudes and medium properties are considered only to be functions of (Z,T).
The wavetrain will then retain an x-depend-
ence only in the phase through the horizontal wave number K.
The resulting theory corresponds to a time-dependent WKB
approximation for finite amplitude gravity waves.
Even though the x-dependence in the wavetrain amplitudes is ignored, the equations remain nonlinear and
numerical computations are required in even the simplist of
cases.
However, the two-dimensional problem has been reduced
to a one-dimensional one resulting in a considerable reduction
in computation effort.
B. Applications of the Method of Characteristics to
Low-Frequency Gravity Waves -'- Some Analytical Results
Let us examine more closely the time-dependent, finite
amplitude WKB
approximation for the case of very low fre-
quency gravity waves, that is, when the doppler-shifted
frequency is such that o2 <<
2
2
For algebraic convenience,
let us introduce a mean wind velocity scale v0 and define the
nondimensional variables:
(1.47)
a)
b)
C)
W
va
V0
TS B
13
CD'
-
+d
The governing equations for the low-frequency case
2 < 1) are:
o
(1.48) Z1
(horizontal momentum conservation)
V
(1.(49).V
(1.50)
a)f
where V
2
.v
#
31
-
(energy conservation)
(phase conservation)
' 0
In linear wave propagation theory, V
would be the vertical group velocity for wavetrains moving in
the negative z-direction.
The system (1.48) - (1.50) can be put in matrix form,
defining the nondimensional vector
(1. 51)
(1.52)
\/
on-w(e
_
),'
=0O
+W_
where
v,
.. n zv4>)
(1.53)
-- Z v6s
0 O9
V0
OI
0
J
Then there will exist a set of three characteristic
curves c , i = 1, 2, 3, defining functions 4i(ZT) = 0, which
are given from the differential equations:
(1. 54)
dZ
i14".
.
L
C
where X
(1.55)
are the roots of the deterministic equation:
(I
.
det[&-X>O-
=
identity matrix)
The eigenvalues, with A given in (1.53) are:
(1.56)
0
a)
V +2.twI
-+
b)
3V
c)
--zv4~f' = V - 21WI
and the corresponding left eigenvectors are:
(1. 57)
a)
.
b
b)
2
2v
[',24
c)
We see that all the eigenvalues are real so the system
is totally hyperbolic in character.
The system can then be
put in canonical form such that in each equation all quantities
are differentiated along the same characteristic direction.
(1.59)
)g
--
(1. 58)
Z
+
+ V.,+ 2V,,
YA-i
(1. 60)
zVo VT ~
-- +-2V
"zv
"']1
0
(5-I±zV
)
PZ+ ve+
2
(VIj~
4
2 V+
, -zVF-FV
20
The details are still quite complicated at this point;
however, two important observations about the system can be
made.
First, in the small amplitude limit where jwj
-+
0,
the eigenvalues (1.56) coalesce to a single value V , and the
system reduces immediately to the linear theory result
(1.61)
-
+
This demonstrates that the total wave energy is conserved as it must be, since there can be no wave-wind energy
exchange without nonlinear interactions.
The second immediate result from the canonical formulation is that there exists one exact integral.
Equation
(1.58) can be rewritten as
(1.62)
which can be integrated directly since $,
C, and riare all
known functions of z at T = 0 in an initial value problem.
Making use of this exact integral, the system can be reduced
further to one containing only two dependent variables.
The
remaining equations are still highly nonlinear, however, the
use of the multiple scale approximation has resulted in a considerable simplification of the original Boussinesq system
given in
C.
(1.1)
-
(1.4).
The Limitations of Multiple-Scale Analysis and the
Question of Partial Reflections
The primary assumption of the multiple-scale analysis
is that there exist at least two time-space scales in the
21
problem under examination.
In the foregoing analysis, we
have assumed that the phase fluctuations of the wavetrainwere
much more rapid than the corresponding time and space variations in the wave amplitudes and properties of the medium.
This assumption means that only a minor amount of wave reflection can take place and thus the wave impedance change per
vertical wavelength mus.t be small. If the reflected waves are
of order e compared to the transmitted wave they will be contained in the amplitude equations resulting from the elimination of secular terms-of order E2.
From linear theory we can evaluate the criteria for
strong reflections.
If temperature and density variations
are slow only the wind variations modify the wave impedance,
and strong reflections result if the Richardson number is
less than 1.
In the nonlinear theory one cannot develop such
an explicit constraint, but it is reasonable to expect difficulties with the multiple scale analysis in regions of strong
wind shears.
In Figure 1 we show some results of calcula-
tions using the multiple-scale analysis for a wave-wind interaction in a region where the Richardson number is 0.53, and
compare them with Breeding's more exact numerical calculations.
The multiple-scale analysis retains the proper
qualitative features, but the amplitudes are considerably in
error.
Breeding's calculations showed that a substantial
reflected wave was present as would be expected with a low
Richardson number, and thus much of the wave energy was turned
back before it interacted with the mean wind.
The multiple-
CHANGE IN THE MEAN WIND NEAR THE CRITICAL LEVEL
- CALCULATION METHODS COMPARISON
0
I
3.0
0
2.9
2.8
.10
2.7
2.6
0
o
0,
0
2.5
z
2.4w
2.3
0
RICHARDSON NUMBER
t = 2190 S.
0--O
=
0.53
N
2.2
MULTIPLE SCALE ANALYSIS
2.0
FIRST AND ZERO HARMONIC
0--
R. BREEDING NUMERICAL
2.0
-1.4
FIGURE
-1.2
-1.0
1
CHANGE IN MEAN WIND IN M/S
-. 8
-. 6
-. 4
-. 2
0
+.2
23
scale analysis which did not properly account for reflected
waves thus gave too large a mean wind modification, and the
discrepancy would continue to grow with time.
When the Richardson number is large, the conditions
necessary for justifying the multiple-scale analysis will be
upheld and then the methods described above can yield good
approximations with a considerable reduction in computational
effort.
Unfortunately, in our jet stream studies, the areas
of greatest interest are those with the lowest Richardson
number, and these analysis methods will fail to give us
adequate representations of the actual behavior of these
regions.
Thus, we are led to find a different approach to
these problems, which is taken up in the next section.
II.
HORIZONTAL FOURIER EXPANSION ANALYSIS
A.
Development of the Governing Equations
In the previous section the multiple-scale approxi-
mation was used to analyze qualitative features of the nonlinear gravity wave - jet stream interaction.
While this
method allows certain problems to be attacked by analytical
means the approximation appears inadequate for obtaining
detailed results in several physical situations.
In particu-
lar, the important problem of interpretation of gravity wave
pressure fluctuations observed at the ground in terms of
possible jet stream source parameters appears to lie beyond
the scope of the multiple-scale approach.
The key to any attempt at inverting gravity wave
pressure fluctuation measurements involves the development of
an efficient, accurate method of solving the nonlinear
hydrodynamic problem under a wide range of initial conditions.
A direct approach would be to use a full nonlinear numerical
model such as that of R. Breeding.
Such a study involves the
use of two-space and one time coordinate finite difference
grid, and while quite general in application, can require a
large computation effort.
An examination of R. Breeding's work indicates that
there is a useful approximation which can be employed to
greatly reduce computation effott while yielding quite satisfactory results under realistic conditions.
The above numerical integration scheme requires the
application of cyclical horizontal boundary conditions, hence
24
the dependent field variables are required to be periodic over
some distance, L, in the horizontal direction.
Therefore, the
velocity, pressure, and density field can be expanded in a
Fourier series in x, where the amplitudes are functions of
(z, t)
only:
*00
(Z t) +
-iky1X
A (Fj
*k
--
J
io()e
(2.3)
net + 0±
(2.4)40,Lt
e
where K =
L ._
Substituting the above expressions for the field vari-
ables in the Boussinesq fluid equations (1.1) - (1.4) and
equating Fourier coefficients we have:
for n = 0,
(2.5)
(tko
(2.6
+
(2.7)+
(2.8)
+
(A*tp
+(
(iOLQ~1i~=
A
'ZZ
0
LoIo+
A-
Z WID
-0 "a
a--"
one
-O
02
3
26
(2.9)
*
-6t
-K
Pl +2.uOLw
'tre
f00
+=2
(2.11)
L.4>, +u,
ok +W
+ 0-0
*-a0C
+2 ;.r WFW~
[P-, + 2wow,
+
+iA*
kAOW, +-IAIWO
-0i K-kAW, 4-U,W
(2.10)
(2.12)
+ 2
4> l-Z LAP~ + LA
+.(w d>,,,++e
<r
<r 3
-Smi
olIw
-dbcux +
and for n > 2.
(2.13)
-- ik
2[
4-2
1t-
e(As
L41 +T
+
ULtW)+
+a+w
Ice-I
= 0
LA P+ v
+ *-W
(2.14)
+ (Ap
y
opt*
)]+- 12
+1 2 ::
Wr
-i
mj.m.
Ak
~
+~3 z OVYM
2
c).
-1- 4)"
rv
4-1. 4,y+ ---
..(y
(2.15)
+0
"Ow, + 4WO) +
.
4)II J+
llbz
*
Loos, +
(+ p
v
>
*p
(w~ve~
27
(2.16)
0.
.
In addition, there are complex conjugate equations for
n > 1. A similar development, including compressibility
effects (other than through the BrUnt period), was used by
Boer (1970) to examine shear flow interaction with a continuous sinusoidal gravity wave source produced by forced motion
on the lower boundary.
At this point no approximation has
been made, given that cyclical boundary conditions are
involved.
There has also been no savings in computation
effort at this step, for if computations are made using N
coefficients, the method is equivalent to a finite difference
net using 2N horizontal points.
However, R. Breeding's calcu-
lations have shown that if the source structure has a dominant
horizontal wavelength X
= 2ir/k, then under most conditions
the higher harmonic coefficients corresponding to n = 2, 3,...
etc., will generally have much smaller amplitudes than the
fundamental corresponding to n = 1.
For example, Figure 2
shows the production of second harmonics due to nonlinearities
using a source containing only the fundamental as calculated
by R. Breeding (1971).
Even for a Richardson number of 0.53,
the second harmonic amplitude seldom reaches 15% of the fundamental and the third harmonic will be correspondingly less.
Therefore, a good approximation can be obtained by retaining
on the first few Fourier coefficients with a considerable
reduction in computation effort.
The simplist approximation available is to consider
the interaction between the mean horizontal flow and the first
2.0
1.0
-
0.5 C.
0.2 -
1
<
0.1 -D
H
I
B
A
0.05 -
o
H
O~
0.02 -
A
.
*
E
D
D
D
D
G
0.01
B
U-0.005
00
-/
0.002
G
0.001
*
A - REFERENCE, Ri
B -Ri = 0.13
A
=0.53
C - Ri = 0.13, NEAR FIELD
0.0005 -Do
D -
Ri = 2.12
E - NEGATIVE SHEAR
*G
G - SMALL AMPLITUDE
H - LARGE AMPLITUDE
I -SOURCE
AT zc
0.0002
0.0001
0.00005
0.00002
0
1000
2000
TIME (s)
FIGURE
2
S/F, THE ABSOLUTE VALUE OF THE RATIO OF THE SECOND
HARMONIC IN u, AS A FUNCTION ABOUT 100m ABOVE zc
(FROM BREEDING (1971))
3000
harmonic wave field.
In this case, the governing equations
become:
(2.17)
-I-
(2.41) (2.1)
b
(2.20)
-
IT, + 2Lko L 1 +
MOM*
-6t
'6 4
(2.21)+
If u0 is constrained to be independent of time and
equal to the initial mean flow, the basic equations oflinear
gravity wave theory result.
Thus the system of equations
(2.17) - (2.21) can be interpreted in a very simple manner.
The gravity waves follow the linear theory for an atmosphere
which is slowly varying in time.
The atmospheric variations
in time result from one nonlinear equation (2.17).
The time
variations of the mean wind have the effect of preventing the
catastrophic build-up of the wave amplitude near the critical
level that linear theory predicts if the atmosphere is considered time invariant.
This result was also seen in the
multiple-scale analysis.
In this present analysis, however,
we clearly see the failing of the multiple-scale approximation.
30
The vertical variations of the mean wind can alter the linear
wave impedance, and thus reflected waves can be set up, and
these are not included in the multiple-scale approach which
is a WKB
type of analysis.
For linear waves in a constant
mean wind shear, the relative change of wave impedance per
radian of vertical wavenumber if 1/R.1 2 and is independent of
actual wavelength.
Thus the shortening of the vertical wave-
length as one approaches the critical level does not help
justify a WKB~ approximation, and for Richardson numbers
close to or less than 1, large errors result if the reflected
wave energy is
ignored.
Again referring to Figure 1, we show comparisons of
the results of various computational methods on predicting
the mean wind variations caused by wave-wind interaction near
the critical level.
Solutions to equations (2.17) - (2.21)
give excellent agreement with the finite difference solution
of the full nonlinear equations.
Again we note that the multi-
ple scale analysis has overestimated the effect, since the
wave reduction due to partial reflections was not taken into
account.
B.
Stream Function - Vorticity Formulation
The numerical integration of the fluid equations for
incompressible Boussinesq fluids requires the reformulation
of the problem in terms of a scalar stream function $, and a
vector vorticity X, rather than the standard Eulerian velocity and pressure fields.
Since the velocity field in an
incompressible or Boussinesq fluid is divergence-free, one
introduces 9 and X in terms of the horizontal fluid velocity
u and vertical fluid velocity w as:
-
.
(2.22)
-
W
and
(2.23)
In the case of only two velocity components, X has
only one component and the resulting stream function/vorticity equation is scalar:
(2.24)
---
t+
--
=
-
where X = YA.
Finally, the equation of vorticity transport is developed by taking the curl of the momentum transport equation.
This operation eliminates the pressure explicitly from the
governing equations.
This is beneficial for numerical exam-
ination of divergence - free flows, where the time derivative
of the pressure is not readily available.
Employing a first harmonic vorticity X
and stream
function $ 1 , the basic system truncated at the first harmonic
becomes:
(2.25)
(2.26)
(2.27)
(2.28)
-
Ao \ 4
* UO)
K
+0
-
=
(2.29)
=-
where
The specification of the boundary condition at the
ground, z
=
0, is straightforward since $l must be a constant
on a rigid surface.
Further, since $l is directly propor-
tional to the vertical velocity w1 because of the Fourier
expansion in x, then $1 = 0 on z - 0. The condition at the
upper boundary z
=
H, requires more care.
Ideally, an out-
ward radiation condition would be imposed; however, the use
of a steady-state radiation condition will cause errors in a
transient calculation; and the addition of a viscous region
as considered by Yanowitch (1967) and Houghten and Jones
(1968), will be computationally inefficient since an additional grid region several wavelengths in dimension must be
included.
Fortunately, the existence of large Richardson num-
ber shears in the upper jet stream structure resolves these
upper boundary condition difficulties.
Numerical results have
confirmed the calculations of J. Booker and F. Bretherton (1967)
that for large Richardson number shears, internal gravit waves
as
are attenuated by approximately a factor exp [-21(Rthey pass through a critical level.
For the case of R
=
4.8,
as per Figure 3, up-going gravity wave amplitudes would be
attenuated by a factor of about 10-2.
Since very little wave
energy will penetrate the upper shear, the impqsition of a rigid
boundary condition above the upper shear will not give rise to
spurious reflections.
Subsequent numerical results have sub-
stantiated this assumption.
At the upper boundary the appropri-
ate boundary value of the wave stream function is $l
-0 at
z
-
H.
III.
NUMERICAL EXPERIMENTS:
WAVE-WIND INTERACTIONS FOR
TRANSIENT GRAVITY WAVE SOURCES
A.
Computational Procedure
In this section we wish to examine through a series
of numerical experiments the nonlinear interaction of a
single harmonic transient gravity wave source with simple
models of the jet stream.
The purpose is not to formally
invert the observed ground pressure fluctuation measurements,
but to examine the qualitative features of the interaction in
terms of the vertical distribution and strength of the source
and initial Richardson number in the source region.
Of
particular interest is the wave-wind interaction in the presence of strong shear (Ri<0.5), since observed gravity waves
often appear to originate from regions of low Richardson
number.
1.
The Finite Difference Equations
The equations describing the wave-wind interaction
(2.25) - (2.29) contain nonlinear coupling terms so that a
numerical approach is dictated.
The finite difference approx-
imation employed here is the so-called "leap-frog" method,
which is an explicit scheme using spatial and temporal
centered differencing.
vertical spacing be 6z.
Let the time spacing be 6t and the
Then for notation purposes, define
(3.1)V
33
34
LX o
{44
The governing finite difference Iapproximations to
equations (2.17) through (2.21) are:
(3.2)
(~3. 3)
~~o~~~
%
ik
Ti
r
[p
~VX~
.. L[
..L1
~~
VOt
P~[Lcj4e
01y.
a-
V%*
V
+
%m1
~n4I
The last equation is solved at the end of each
iteration as a boundary value problem with
=.
-0
where j = 1 and j = J MAX are the end points of the finite
difference grid.
Two other approaches to solving the stream
function-vorticity equation involve an integral method using
Green's functions and a transform method using Fast Fourier
Transform (FFT) techniques.
Solution of the tridiagonal
system (3.6) and the FFT method are about equal in computing
time, whereas the Green's function method is much slower.
The
FFT method is, in principle, more accurate than the secondorder difference approach given in equation (3.6).
However,
a fourth-order differencing scheme,
(3.7)
+1-
12
gives much better accuracy with only a slight increase in
computing cost over the use of the FFT.
All three methods
were tested in practice and gave quite similar results.
In
the case of very low Richardson number shears, the fourthorder scheme was used because of the short wavelengths
involved.
2.
Convergence,
Stability and Grid Dispersion
Of fundamental importance in applying finite difference approximations to the solution of partial differential
equations are the questions of convergence, numerical stability and accuracy.
When dealing with wave propagation phenom-
ena, lack of accuracy of the solution usually manifests
itself as grid dispersion.
Unfortunately, analysis of these
effects is limited to a linear, point-wise approach.
The
global analysis of a given finite difference scheme is fully
as difficult as solving the originally partial differential
equation.
To examine the point-wise characteristics of the
difference equations it is useful to derive the governing
finite difference dispersion relation.
analysis, let
;
constant and
'
Since this is a local
constant.
This is not a limitation in terms of the derived stability
Consider plane
criteria, but greatly simplifies the algebra.
wave solutions for the field variables as:
-\
(3.8)
~=cte.
-%
A'
= (wstW t +
where
(LoS-)
is the phase, and
and (.q[e) are the normalized frequency and vertical
Substituting into equations (3.4),
wave-numbers respectively.
(3.5) and (3.6) and defining,
(3.9)
St
1 2EZ
47=
gives, in matrix form
(3.10)
O0
A
o
-t
<1
c>ng
(et -- coS t)
tO
Setting the determinant in (3.10) to zero yields the dispersion
relation,
(3.11)
Sirn (00$t)
= K(Q L:.
In the limit that 9't, 9' -% 0
(3.12)
LOk WA
4 (-Cos
)
(3.11) becomes
KL
which is exactly the dispersion relation for the continuum
37
equations.
Convergence is therefore established, at least on
a point-wise basis.
The wave solutions described by the finite difference
dispersion relation will, in general, differ from those of the
continuum case.
In certain instances, completely extraneous
wave modes arise which correspond to complex frequencies.
Since these modes will occur as complex conjugate pairs, the
exponentially growing mode leads to numerical instability.
From equation (3.11), unconditional stability requires that
Sivi(tot)
I
for all wavenumbers a.
will occur at cos a = 1 for the plus root:
SiV1(jst)
(3. 13)
The maximum value of
Mo.A.-[S iki (LOS+3=t-)('v41
The requirement on the time step 6t for unconditional local
stability is then
(3.14)
St
:
KZ +S2g)
Grid dispersion in a finite difference scheme arises
because the stable modes of (3.11) corresponding to a wavenumber-frequency pair are not equal to the same wavenumberfrequency pair in (3.12) for finite 6t, 6z.
Subsequently,
the phase velocity of a given mode will also differ from that
of the continuum, and in general this correspondence will
vary as a function of wavenumber and propagation angle.
Therefore, superimposed on the wave dispersion of the continuum system will be a propagation-angle dependent numerical
or grid dispersion.
The loss of accuracy resulting from grid
dispersion is inherent in all finite difference approximations
38
and can only be minimized, with the lower bound on grid
spacing usually determined by computation cost.
All numeri-
cal computations were done with a step size of 6z = 25m in
regions where short wavelengths are expected, such as near
critical levels and source regions.
Reasonable accuracy is
then expected for wavelengths greater than about 100m.
with
only moderate to poor resolution for shorter wavelengths.
B.
Basic Jet Stream and Gravity Wave Source Models
For use in the numerical experiments a basic jet
stream mean flow model was employed.
The initial jet stream
velocity profile was assumed to be uniform in the horizontal
plane with a vertical wind profile.
The direction of the
mean wind was chosen to coincide with the x-axis.
The general
model is shown in Figure 3, which is divided into three
regions.
Region II is taken to be 2.0 km wide centered about
the source.
The source subregion itself will range in width
from about 125 m to 550 m.
to a height of 4.0 km.
Region I extends from the ground
The remainder of the jet and compu-
tational grid is contained in Region III with an initial jet
core speed of 50 m/s.
Except for the source subregion, the
Richardson number in the lower and upper shears is fixed for
all models at R.
II
=
0.925 and R.
IIII
=
4.80 respectively.
The source subregion is centered at 5.0 km and the width and
initial Richardson number are specified for each model.
All
shears are initially taken to be linear.
The basic structure of the jet stream model was
developed from examination of typical Weather Bureau balloon
39
MODEL JET STREAM PROFILE
20
A
15
Ri = 4.80
z
(-9
JET STREAM
CORE
Lu
Ri
0.925
U
5 t-
SOURCE SUBREGION OF SPECIFIED
WIDTH AND RICHARDSON NUMBER
_
__
MEAN HORIZONTAL WIND SPEED IN M/S
FIGURE
3
SCHEMATIC OF MODEL JET STREAM PROFILE
INCLUDING REGIONAL AND SOURCE SUBREGIONAL
CONFIGURATION
I
m
40
measurements over Nantucket Island.
In general, the balloon
measurements indicate the jet stream velocity profile to be
quite symmetrical; however, because of the change in the
temperature lapse rate above the jet core, a higher Richardson
number results in the upper sheared region.
Therefore, in
order to use a constant Brunt frequency model in the calculations, the shear in the upper region has been decreased so as
to yield an appropriate value for the Richardson number.
The simplest source models which will retain the
essentials of the nonlinear interaction between finite amplitude gravity waves and the mean jet stream flow are those
containing only the first harmonic component of the general
horizontal Fourier expansion described in Section II.
The
general form of the numerical experiments is to specify a
gravity wave source structure implanted in the model jet
stream flow as an initial condition.
The behavior of the
model is then governed by the finite difference equations
(3.2) - (3.6), which describe the interaction of the zeroth
and first harmonic terms.
The development of theoretical atmospheric gravity
wave source models represents a very difficult problem in its
own right.
A recently developed approach to gravity wave
source generation by wave-wave interaction will be discussed
in Section IV.
However, at this point a more heuristic
approach will be taken.
Although the specific choice of a
model is admittedly arbitrary, there are constraints on
physical grounds and in terms of linear theory behavior.
41
Gravity wave-wind interactions essentially only exist around
a narrow region about the critical level.
Therefore it is
reasonable to locate the source at a critical level and
Linear
require it to rapidly decay away with distance.
theory shows that the horizontal wave-wind velocity has
opposite signs on different of the critical level.
Further,
the vertical wave-wind velocity must go to zero at the critical level while the horizontal wave-wind velocity becomes
discontinuous with infinite amplitude.
Any source model
should approach these limits as the amplitude of vertical
velocity and the source width go to zero for a source of
fixed total energy.
Under these constraints, a simple source model is
chosen with a Gaussian distributed stream function:
-t=o)
(3.15)
= 2A e,
Cos ( ,
The associated velocity field is:
(3.16)
WtxA2,t=O)-z
ce
Sin kX
Z
(3.17)
')
X),t=0)Z
=or
q
-L
C
If we consider the limit as A+O and a->oo,
the con-
straint of constant total source energy requires A2a
straint, then w
a-
and uca
k-
=
con-
(z-z ), so that linear
theory behavior at the critical level is obtained.
Wave and Wind Energy Calculations
42
In a Boussinesq fluid, the conservation of energy is
given as
tE + ?2
(3.18)
LE
+
(.
where u and w are the total horizontal and vertical velocities, and E is the total energy density.
terms of the field variables, u, w,
(3.19)
E
=5
Expressing E in
and P, we have:
+ /2 1-4
+
(recalling that P = pressure/mean density).
If we consider the total Eulerian field to be composed of a mean flow and finite amplitude single harmonic
wave, the total energy density averaged over a horizontal
wavelength is
(3.20) < 1E
(o?'t) + Lk'Lk +V IVNJ4
+ C
The equation of energy conservation for this system is obtained
from averaging (3.18):
(3.21)
,.....ow
or equivalently,
(3.22)
(E>>
+r
Lkuu
+f,'W+P
+
W)=
One can easily demonstrate that the energy conservation
form developed directly from the zero and first harmonic
expansion equations yield an identical result.
Integrating
43
(3.21) over z from 0 to H gives
(3.23)
< WE).
d E- <E%
01
0 and z - H, then
If we assume rigid surfaces to exist at z
=
the vertical velocity must vanish there.
Therefore,
14
(3.24)
d
E
0
and the total integrated energy per wavelength must be a
The behavior of
constant over time in such a closed system.
the computed total system energy as a function of time provides one of the constraints on the global behavior of the
finite difference scheme.
It will be useful in the analysis of the numerical
experiments to examine the behavior of wave and wind components of the total energy as a function of time in each of the
three regions of the model.
Since the total kinetic energy of the mean wind is
very large with respect to the other components of the total
energy, it is useful to redefine the total system energy
excluding the total wind kinetic energy at t = 0.
Changes in
the wind energy will then be reflected by alterations in the
mean flow caused by wave-wind interactions.
Under this defin-
ition, the total energy of the system will be equal to the
total source energy at t = 0. Using this definition, the
wave and wind energy densities are from (3.20).
(3.25)
c AAit
4
OE
z(L(,V
44
(3.26) <E>
I
,
)
where
4-/
*
+2
and
Z ,
L(AZ-ft)
is
the time-
varying change in the mean wind.
To determine the total energy input to each model we
need the total source energy.
from the critical level as
Since the energy decays away
6
and the source width
is at most a few hundred meters, the integration interval
0 to H
=
20 km can be extremely well approximated by (
In this case the source energy at t
(3.27)
E,
=
0 is
J'(crtz
<Egr~Po
The total system energy and the source width parameter a can
be used as input parameters to completely describe the source
model.
C.
Transient Behavior of Finite Amplitude Gravity
Wave Sources in Low Richardson Number Shears
The gravity wave source and jet stream models to be
examined numerically will have the following fixed and variable parameters.
Fixed parameters for all models:
(1) Jet stream mean flow structure excepting the
source subregion:
(a)
upper shear
(b) lower shear
R
R.
3.
(c) jet core speed
=
4.800
0.925
50 m/s
(2) Source centered at 5.0 km.
45
(3) Wind speed at source initially 25 m/s.
(4) BrUnt period
300 s.
(5) Horizontal wavelength
10 km.
Variable parameters:
(1) Total source energy E
(2)
at t = 0.
Source width at t = 0.
(3) Richardson number of source subregion at t
Specific models given in figures:
MODEL A at t = 0
R. = 0.200
width = 318 m
peak wave velocities
horizontal = 0.085 m/s
vertical
= 0.010 m/s
MODEL B at t = 0
R.= 0.250
width = 318 m
peak wave velocities
horizontal = 0.85 m/s
vertical
= 0.10 m/s
MODEL C at t = 0
R. = 0.500
width = 318 m
peak wave velocities
horizontal = 0.85 m/s
vertical = 0.10 m/s
=
0.
46
MODEL D at t = 0
R.
=
0.500
width = 136.5 m
peak wave velocities
horizontal = 1.30 m/s
vertical
=
.066 m/s
MODEL E at t = 0
R.
=
0.500
width = 273 m
peak wave velocities
horizontal = 0.92 m/s
vertical
= 0.092 m/s
MODEL F at t = 0
R. = 0.500
width = 546 m
peak wave velocities
horizontal = 0.64 m/s
vertical
= 0.13 m/s
The phenomena observed during the wave-jet interaction
will be discussed in terms of the time-energy curves given in
Figures 4 through 7. The detailed vertical distribution of
wave and wind velocities in regions I and II for Model A are
given as a function of time in Appendix C.
The energies shown
refer to the integrated energy densities within each of the
three regions shown in Figure 3. The time-energy curves give
spatially integrated values for both the wave motion and changes
in the mean wind structure, whereas the time-net energy curves
refer to the total system energy integrated over each region.
-.-.-.-.-.-.-.-.
--
20
-
---
-
WAVE ENERGY -
REGION I1
WAVE ENERGY -
REGION I
WIND ENERGY -
REGION 11
REGION III
WIND ENERGY
-
10
.
-6
.0
--
-10
-20
1000
0
FIGURE
2000
t,sec.
3000
4000
4
TIME ENERGY CURVES FOR MODEL A
-4
#
40
o
0
0
REGION I
I REGION 11
x x x REGION III
E/EO
0
1000
FIGURE 5
TIME - NET ENERGY CURVES FOR MODEL A
2000
T, SEC
3000
4000
49
MODEL B R = .250
*o.
*Og
-S.
E
-S.
S..
-
%
-
-
-
-
__
~
-
a,
\
\\
-10
-20
-
NK- ~
-
a-
L.
MODEL C R= .500
E
0 a
-0-
-
-
-10
-20
1000
t, sec.
FIGURE
6
TIME ENERGY CURVES - RICHARDSON NUMBER EFFECTS
2000
50
MODEL D WIDTH = 136.5 m
E
Eo
N
-5
a-
MODEL E WIDTH = 273.0 m
5
E
~
0
-
-
..
-5
L-
MODEL F WIDTH = 546.1 m
" \
-
-5
-
'p
,~
wo-
.. -
-
I
1000
2000
t, sec.
FIGURE
7
TIME ENERGY CURVES - SOURCE WIDTH EFFECTS
51
All energies shown are normalized to the total integrated
input source energy, E , at t = 0.Wind energies then refer to
changes from the mean flow state at t = 0.
The boundary
conditions of the jet model require that the total system
energy integrated over the entire model remain constant and
provides an estimate of cumulative truncation effects of the
finite difference solution.
A maximum deviation of eight per-
cent from the condition of constant total system energy was
observed after 400 time steps for the case of R. = .200.
The
total system energy typically oscillated about a slowly
decreasing mean value.
The period of the oscillations was
about 100 time steps with an amplitude of less than
0.01E .
The mean value decreased as a function of the number of time
steps as very short wavelength disturbances are dissipated
by the finite difference scheme.
The total system energy
after 200 time steps (= 2000 s in model time) ranged from
E = .995E
for R. = 0.925 to E = .97E
for R.
=
0.200.
Results:
The initial interaction in the source region extracts
considerable wave energy at the expense of the wind.
The
amount of wave amplification increases dramatically with
decreasing Richardson number as seen in the time-energy
curves, but is observed in all cases even when the Richardson
number was greater than 0.25.
As the wave energy in the
source region nears a maximum, substantial gravity wave
energy is being radiated away from the region as observed
from the increase in wave energy in region I. Approximately
equal amounts are radiated up the jet and toward the surface.
Waves traveling
up the jet transport negative momentum and
will decrease the wind in the region about the topside critical level.
Waves traveling*: toward the surface are carrying
positive momentum.
Until the return to the source region of ground
reflected wave energy, the net energy in the source region
declined slightly but remains near its initial value of E/E
- 1.0.
The two-way travel time from the source critical level
for the surface reflection can be shown to be approximately
700 s on the basis of linear calculations, and at about this
time the net energy in the source region is rising rapidly.
Since the ground reflected waves are moving faster than the
wind as they approach the critical level, they are carrying
positive momentum and will increase the mean wind and net
energy in the source region on being absorbed.
Waves traveling
up the jet are moving slower than
the wind as they approach the topside critical level so
aE/Dt < 0 in this region.
If no secondary sources are gener-
ated and no additional gravity waves are emitted from the
source region, the.net energy in Region III will approach a
negative constant as wave motions are converted to mean flow
deformations.
> .25 this was the observed process,
For R.
3.
however at lower Richardson numbers secondary sources do
result with subsequent gravity wave radiation into Regions I
and 1II.
For Model A this process is observed in Figure 5
with slight increases in Region I and corresponding decreases
in Region III being seen at t
-
1400 a. Additional secondary
53
radiation occurs at t = 2100 and t = 3400, although the system
remains stable with each subsequent radiation process being
less energetic than its predecessor.
The radiation, reflec-
tion and reabsorption processes will continue until virtually
all wave motion has been converted to mean flow deformations
through the efficient gravity wave - mean flow coupling mechanism at the source and topside critical levels.
Of particular interest is the amount of energy transport provided by the gravity wave coupling of the topside
and source region critical levels.
The amount of net energy
transport from the critical level at the top of the jet to
the source region critical level is shown in Figure 9 and is
a strong function of the initial source subregion Richardson
number.
For the case of R
= 0.200, 16 energy units per unit
of input have been transported from the top of the jet to the
source region.
Even for a Richardson number of 0.500, the
energy transport is three times that of the source input.
For a fixed initial total source energy, the effect of
decreasing the width is to increase the efficiency of the
energy transport process and lengthen the time scale of the
interaction.
The time scale of the interaction can be seen
from Figure 6 to be directly proportional to 1/width.
For
instance, the wave energy maximum in the source region occurs
at 900, 450 and 225 s for corresponding widths of 136.5,
273.0 and 546.1 m.
The effect of absolute wave amplitudes on the system
behavior is approximately linear.
Shown in Figure 8 is the
source region wave amplification as a function of initial
source input energy for a fixed width of 318 m.. The net
energy transport also shows a slight, but definite increase
with increasing source energy input.
However, the ampli-
tude behavior is essentially linear over decade variations in
input energy.
For very large source wave amplitudes (greater
than 10 m/s), rapid and large variations in the mean wind
can occur with subsequent Richardson number reduction to
below 0.10 and generation of short vertical wavelength disturbances.
Neither of these effects can be dealt with accu-
rately by the current numerical procedure and probably
invalidate the initial assumption of a single component - mean
flow interaction model.
The maximum wave amplitudes success-
fully modelled were observed to be less than 8 m/s in the
source region.
Of fundamental importance is the magnitude of wave
velocities in the source region necessary to produce the peak
to peak pressure fluctuations observed at the ground.
Typi-
cal measurements of jet stream associated gravity waves in
eastern Massachusetts show peak to peak pressure amplitude
variations of 0.1 to 0.5 millibars, although disturbances
occasionally 1.0 mb or greater in amplitude are observed.
Shown in Figure 10 are the calculated peak to peak ground
pressure amplitudes as a function of the maximum rms wave
velocity in the source region for models with varying initial
source subregion Richardson numbers.
The observed 0.2-0.5 mb
pressure amplitudes are being produced by source region wave
velocities of 1-2 m/s which are well within the limitations
R =.200
4.0 F-
R = .240
-%-.
---
3.0
x
R = .500
2.0
I-'
\:J
R
1.0
FIGURE
.001
8
SOURCE REGION WAVE ENERGY:
AMPLIFICATION FACTOR vs.
MEAN INITIAL ENERGY DENSITY
= .925
0.10
0.01
m/sec.
1.00
0
10.0
56
E/EO
0.56/R;
10.0
0.5
1.0
0
1
0.33
0.25
0.2
3
4
5
2
1/Ri
E= TOTAL SOURCE ENERGY AT T = 0
FIGURE 9
NET ENERGY TRANSPORT BY GRAVITY WAVES AS A FUNCTION
OF SOURCE SUBREGION RICHARDSON NUMBER
1.0
INCREASING R
0.1
0.925
0.500
0.240
0.200
0.01
0.1
I
I
I
i
1.0
MAXIMUM RMS SOURCE REGION WAVE VELOCITY IN M/S
FIGURE 10
PEAK-PEAK GROUND PRESSURE FLUCTUATIONS IN MB
10.0
58
of the model.
The increase in transmission efficiency for
higher Richardson number is due to the relative decrease in
the partial reflections in the source region resulting from
reduced impedance change per wavelength.
IV.
THE DEVELOPMENT OF JET STREAM GRAVITY WAVE SOURCES BY
WAVE-WAVE INTERACTIONS
A.
Introductory Remarks
The results developed thus far describe the propaga-
tion effects and energy transport in the jet stream due to
single harmonic finite amplitude gravity wave sources.
The
question of the origin of natural jet stream sources has been
left unanswered.
In this section we will discuss a possible
mechanism for the generation of such gravity wave sources.
Observational information shows a strong correlation
between horizontal phase velocities of gravity wave pressure
fluctuations at the ground and jet stream velocities in
regions of low Richardson number (Claerbout, 1967).
A natural
mechanism would seem to be the amplification of small disturbances due to dynamic instability in regions of the jet
stream where the Richardson number falls below 0.25.
A large
body of literature exists on the instability of shear flows to
small perturbations within the appropriate range of horizontal
wavenumbers and Richardson numbers (see References).
However,
the wavenumber dependence of instability is such that exponentially growing waves in the jet stream source regions will
have periods less than the BrUnt period and are essentially
trapped within the jet stream.
Typical power spectra of
ground pressure fluctuations show a sharp cutoff for periods
less than the Brunt period (Fig.ll).
The dominant observed
periods correspond to waves which are predicted by linear
59
60
Odb
10db I-
Odb
x%
20db
-
,- %
SMOOTHED WESTON
10ildb
GRAVITY WAVE SPECTRA
0-X- - - X-- COMPUTED SUBHARMONIC
SPECTRA
BRUNT PERIOD
30db
3.0
22.8
2.5
2.0
1.5
15.2
FIGURE 11
SMOOTHED TYPICAL GRAVITY WAVE SPECTRA (NORMALIZED)
OBSERVED AT WESTON, MASS. AND THEORETICAL VALUES FROM
SUBHARMONIC GENERATION.
1.0
T/TB
7.6
T, min.
61
stability theory to be stable in source regions even if the
Richardson number is less than 0.25.
These observational
difficulties can be resolved through a wave-wave interaction
mechanism.
B, ' Brief Review of Linear Stability Theory
The infinitesimal stability of inviscid, parallel,
stratified shear flows is an area of extensive study in fluid
mechanics and applied mathematics.
Using a single Fourier
component of the vertical perturbation velocity w(z), with
horizontal wavenumber k, the linear stability of the flow is
governed by the Taylor-Goldstein equation in the Boussinesq
approximation,
(4,)
....- +
[f---
--
where v(z) is the mean flow and C the horizontal phase speed.
Together with appropriate boundary conditions, it defines an
eigenvalue problem for C given K, or vice-versa, for a specified mean flow v(z) and BrUnt frequency 0B'
Often of particular importance is the calculation of
"neutral curves," which for a given value of Re (C) divide the
Im(C)-Richardson number plane into regions of stability (Im(C)
> 0, say) and instability (Im(C)
matical subtleties.
< 0, say).
There are mathe-
However, for asymmetric flows, it can
be shown that the neutral curves separate simply-connected
regions
(Miles, 1961, 1963).
Separate flows have been
examined in the literature (Collyer (1970), Drazin and Howard
(1966), Hazel (1972), Maslowe and Kelly (1971); however, for
62
the sake of explicitness the remainder of the chapter will
concentrate on the hyperbolic tangent mean velocity profie.
Stability of the Hyperbolic Tangent Mean Wind Profile
with Constant Brunt Frequency
Let the mean wind v(z) be written in scaled terms as
v(z) = vof7
(4.2)
+ tanh(z/d)]
where d is a vertical scale length.
We can then introduce
further scaling,
(4.3)
E-,; a E Kd
0
and the Richardson number dependence upon z is
(4.4)
R.(z) = J
1
o
cosh 4 (z/d),
~2
where Jo0
2
is the minimum value of R1(z) occurring at
ve
z = 0.
IzJI.+
The appropriate boundary conditions are w(z)
-+
0 as
All eigenvalues have the same real part, R (C)
=
1,
and the neutral curve [Im(C) = 0], was first obtained by
Drazin (1958) and is given analytically as
(4.5)
Jo
a (1-
a)
The eigenvalues, except on the neutral curve where
C
=
1, are not available analytically and must be computed
numerically (see Appendix A).
The stability diagram for this
flow is given in Figure 12, showing the neutral curve, intermediate eigenvalues at J0 = .15 and .20, all with the curve
63
J
=
2
The latter represents a condition of BrUnt
.
frequency reflection somewhere in the shear for wave with
phase velocities of C = 1 (see Appendix B).
Im(C)
Of course, if
#
0, there will not be total reflection; however, the
attenuation lengths for J0 > a2 are much greater than for
j
< a2.
if
Ci
1, the attenuation lengths are given
approximately by
jo>o(:
0_
d
.
__
(4.6)
Since C2 << 1 we observe that unstable waves are
effectively trapped in the source region, which strongly supports
the observational evidence.
Furthermore, frequently observed
ground pressure fluctuations must represent waves which are
theoretically stable in the source region.
C.
Source Generation by Subharmonic Wave-Wave Interaction
One possible mechanism for generating observed waves
is through a wave-wave interaction involving an unstable mode
of horizontal wavenumber 2K and a stable (observable) wavenumber K.
Laboratory experiments by Browland (1965), Miksad
(1970) and Scotti and Corcos (1972) on free shear layers
indicate
that a growing fundamental mode interacting with the
background noise could generate substantial energy at the
subharmonic frequency.
An examination of the stability
64
diagram (Figure 12) indicates that the subharmonics corresponding to the unstable modes would have periods of about
1.2 to 1.6 times the Brunt period, which fall in the range
of substantial observable gravity wave energy.
To examine the plausability of the subharmonic generation mechanism one could employ the zeroth, first and second
harmonic equations developed in Chapter II using very samll
initial values for the primary and subharmonic waves.
The
unstable mode would first be expected to grow exponentially
at a rate determined from linear theory.
The growth of the
subharmonic is proportional to the amplitude of the primary
so eventually its growth will be very large, thereby altering significantly the primary wave growth, perhaps even
stopping it altogether.
Unfortunately, the finite differ-
ence solution will be very costly because of the small time
steps required to accurately follow the large expected growth
rates,
This problem is indigenous to finite difference simu-
lations of physically unstable systems.
However, an approxi-
mate approach can be taken to qualitatively examine this
generation mechanism.
Let us assume that each wave behaves as it would in
linear theory, excepting that the nonlinear wave-wave interaction terms will produce a time variation in the amplitude.
The horizontal (complex) wave velocities are given then as:
(4.7)
u = Ae
)
Jo = R j (z = o)
= min (R)
0
0.05
0.1
0.15
0.2
0.25
STABLE AND PROPAGATING
.2
.3
.4
.036
a
Jo = a
2
.063
.5
.075
.6
UNSTABLE
AND TRAPPED
.079
.077
.7
.071
.060
.8
STABLE AND TRAPPED
.046
.028
1.0
FIGURE 12
STABILITY DIAGRAM FOR HYPERBOLIC TANGENT PROFILE
AND COMPUTED EIGENVALUES
NEUTRAL CURVE
(Im(c) = o)
66
1.0
.9
Jo = .150
.8
.7
.6
27rC
a
.5
-5
Jo = .200
.4
.3
.2
.1
0 13
.2
.3
.4
.5
.6
.7
FIGURE 13
NORMALIZED GROWTH RATE VS. WAVE NUMBER
FOR UNSTABLE MODES OF THE HYPERBOLIC TANGENT PROFILE
.8
.9
1.0
67
where a = Im(w) for unstable mode, computed for a hyperbolic
tangent shear model with constant Bru*nt frequency.
The non-
linear terms in the horizontal equations of motion are:
-- o
+
-21tKttl (A +
Ntt
'* 1 *
, +L-U
k
(4.8)
Then if we multiply each equation by f
and f2 respectively
and average over the region (R,-R) corresponding to the Brunt
reflection of the unstable mode, we have:
(4.9)
(4.10)
_-
C, A
Z -
2
dat
A2
where
-R
R
e
68
The formulation given in equations (4.9) and (4.10)
are analogous to those used by Thorpe (1966) and Garret (1968)
For
in discussing resonant interactions in stratified media.
the case of homogeneous media, the vertical distribution
functions are simple plane wave solutions and the interaction
integrals can be evaluated by inspection.
A full discussion
of resonant wave-wave interaction for gravity waves in homogeneous media is given by Phillips (1968), with further
developments by Craik (1968), Simmons (1969) and Martin, etal
(1972).
Due to the very restricted phase structure, resonant
gravity wave interactions in homogeneous media require a
triad resonance such that
(4.11)
.
-
and
(4.12)
W.,Z.
=(D.
W3
In stratified shear flow this is not the case, however, due to the possible extensive phase overlap between the
unstable mode and its subharmonic.
In order to estimate the integrals, let us assume
that the significant contributions to the interaction will
occur very near the critical level (which is the same for both
modes).
The solution of ( 4.1) for the normalized vertical
velocity structure for z/d << 1 can then be approximated
for z positive as:
WccZ
(4.13)(c
____P/i
V0
VO
The corresponding forms for the functions f
and f2
are:
.L
(4.14)
come8
L--W
V.
---
1
-U
(I Afz
(4.15)
Using the fact that near -n
(4.16)
-+ Vf
2
C>
C.
and
(4.17)
y
CL
T
we have, defining h(j)
(4.18)
48 A
'1
ci
C--w-L
b4%
-A)
n-
*
,V
+
(4.19)
-Q~
dit
L18
V%
1 L~~ 4
4#)
:~~,M)
(1+
Since the solutions to (4.16) and (4.17) apply only
near the critical level, the upper limit r is difficult
establish.
that
(4,20)
The criterion for use of ( 4.16) and (4.17) is
and since
2.
The integrals cannot be done exactly; however, retaining only the dominant terms (r<< 1) gives
=
(4.21)
(4.22)
.q
=
-
Toc
C
where
e
A
--
E = 4z-' (w/)
If we further scale the time as
(4.23)
=
-r
.53
t
then (4.21) and (4.22) become
(4.24)
j(AlVo)
a
Lr.A/C._
*(Y/s(Bl)1
dc
(4.25)
AC0/vO)
C
s4(AV2YJ
There remains the choice for the upper integration
limit r. A reasonable choice would be to take z corresponding to r to be one-half of the widths used in the subharmonic
transient calculations.
If W is the width, then r is given
directly as
(4.26) v
-
30
s.,
0
For values of TB= 300 s., W = 150Omand V 0
r = .06/JOP
mation.
so that r2 <<
=25
m/s;
o/2 as required in the approxi-
Given an unstable mode corresponding to a (a, Im(c))
71
pair for a specified Richardson number at z = 0 (J ), we can
compute the coupling coefficient C0 .
With the initial values
of A and B specified, A(t) and B(t) can be found by direct
numerical integration of (4.24) and (4.25).
Shown in Figure 14 is the time behavior of the energies for the least stable mode with R.(z = 0) = J
= 0.200.
The eigenvalues of this mode are a = 0.65, Im(C) = 0.044 and
the initial amplitudes were:
JA0 1 = |B| = .004U0 (corres-
ponding to peak horizontal wave velocities of 0.1 m/s for
U0 = 25 m/s).
Also shown is the total energy extracted from
the wind to drive the process.
In this simple model there is
no limit to the energy that can be extracted by the unstable
mode.
Clearly there will be a point where the energy extracted
will have substantially altered the wind, and therefore the
interaction.
The cutoff line shown in the figure corresponds
to an average reduction in the wind of 16%.
The peak amplitude of the subharmonic corresponds to
a velocity of about 5 m/s in the source region.
Using the
parameterized relationship between peak source wave velocity,
Richardson number and ground pressure fluctuations, we would
expect a peak to peak ground pressure of 0.5 mb, which
corresponds very well with observations.
In Figure 11 the
expected peak to peak ground pressure fluctuations are shown
as a function of period as calculated from the peak horizontal subharmonic velocities generated in the simple model
considered.
Overplotted are the averaged observed values.
The simplicity of the model certainly precludes any statement
1.0
-
-
Eprimary/U 2
--
-10 MIS
Esubharmonic/U 0 2
0
00
0
TOTAL SYSTEM/UQ2
R = 0.200
= (kd) =0.65
Ci/Cr = 0.044
VELOCITY MARKS
ASSUME U0 =25 M/S
10-2
0
2
0
E0
1.0 M/S
10-3
00
/o1-
10-4-
0.1 M/
10-5
0
2
4
6
8
t/TB
FIGURE 14
TIME BEHAVIOR OF PRIMARY
AND SUBHARMONIC ENERGIES
10
12
5 M/S
INITIAL VELOCITY MAGNITUDE
X 2 M/S
RATIOS (FOR Uo = 25 M/S)
0---co .004
(.01 M/S)
.004 (0.1 M/S)
X-X
R ; = 0.200
1 M/S
T
TB
.60
.80
TB
T
FIGURE 15
SUBHARMONIC ENERGY SPECTRUM
- MAXIMUM OUTPUT VALUES
f
4)
1.0
74
of the decisiveness of the theory, however, the close similarity of the spectral shape and amplitude do indicate the
plausability of the subharmonic generation mechanism.
Figure
14 shows that the subharmonic waves will be present in the
source region with sufficient amplitudes for interaction with
the wind for about two Brunt periods.
This is the same time
scale observed for the subharmonic-mean wind interaction
studied in Section III.
Additionally, the growth of the
unstable mode is slow relative to the Brunt period and its
amplitude is small enough so that little modification of the
mean wind will occur until substantial subharmonic amplitudes
are reached.
The time scales of the subharmonic-unstable mode
interaction are therefore compatible with those of the subharmonic-mean wind interaction found in the transient gravity
wave source calculations.
As a note of numerical interest, accuracy of the
integrations of (4.24) and (4.25) required a time step 6t such
that 6t < .002T
For a system including the zeroth, first,
and second harmonic, the total computation time would be at
least 20 times as great as that for the zeroth-first harmonic
system for the same vertical grid spacing.
As a final point, although the lowest Richardson
number we have observed on Weather Bureau balloon measurements
is 0.25, the existence of regions for stability with R. < 0.25
seems likely.
The absence of observations of Richardson
numbers as low as 0.20 can be explained on the basis of the
vertical sampling of the mean velocity.
If the entire lower
jet were modeled by a hyperbolic tangent profile with a source
75
velocity Us, Brunt period TB, and a vertical sampling distance
Az, then the lowest observed Richardson number would be
with the sampling parameter Ao given as
2Az
Using the balloon measurements for typical values of Us, B'
and for minimum Richardson number points and a true value of
0.200, the estimated Richardson number would be 0.243 if the
critical level were exactly between the sample points and no
noise was present in the observations.
An error of half a
sample point would give a value of 0.260 for the minimum
Richardson number.
V.
SUMMARY
The first approach developed to study finite amplitude
gravity wave-wind interaction was based on a multiple scale
analysis technique.
The assumptions are that there exist two
time and space scales governing the dynamics of the waves and
wind:
(1) a "fast" scale corresponding to the relatively
rapid temporal-spatial variations of the wave phase, and (2)
a "slow" scale corresponding to the variations in wave amplitude and mean flow.
The analysis is nonlinear in that wave
amplitudes are not considered infinitesimal.
The resulting
theory is equivalent to a finite amplitude WKB approximation.
The limitations of applicability are much like that of linear
WKB theory.
That is, partial reflections are considered to
be second-order effects so that appropriate use of the theory
implies that the change in impedance per vertical wavelength
be small.
For the gravity wave - sheared flow interaction,
the change in impedance per wavelength is proportional to
1/R5,
so the theory is asymptotically correct as R.
will give good results only for R. > 1.
and
If the theory is
applied to lower Richardson numbers, the change in the mean
wind is overestimated since partial reflections are neglected
to first order.
Since observed gravity waves appear to originate
from the least stable portions of the jet stream where R. < 1,
.
a theory incorporating partial reflections as well as finite
amplitude effects was required.
A quasi-linear theory was
developed in section II which made no restrictions on the
phase and amplitude structure of the waves and was well suited
to transient source calculations.
The equations of motion are
those of linear theory except that time changes in the mean
flow due to wave-wind interaction are incorporated through a
momentum transport equation for the mean flow.
The waves then
propagate through a time-varying medium whose time variations
are produced by gradients of the wave Reynold's stress.
theory has certain appealing features.
The
The nonlinear inter-
action takes place in a small region about the critical level
and these changes are directly incorporated.
The induced time
changes in the mean flow near the critical level are included
in equations of motion for the waves, thereby eliminating the
singular nature of the linear theory.
Away from critical
levels, linear theory describes the propagation of gravity
waves quite well and the quasi-linear theory becomes essentially linear there since there is virtually no wave-wind
interaction.
The intuitive aspects of linear theory can
therefore be applied except near critical levels.
From a
computational point of view, the quasi-linear theory is much
more efficient than a full nonlinear calculation as only one
space and time variable are required since one can consider
waves to be harmonically varying in the direction of the mean
flow.
Quite general one-dimensional source and jet models
can be modelled for transient calculations.
Using the quasi-linear theory several numerical
experiments were made using a model jet to examine the transient behavior of finite amplitude gravity wave sources.
The
calculations of the effects of finite sources show that
considerable energy is withdrawn from the wind in the source
region and put into wave energy.
The gravity waves very effi-
ciently couple the critical level at the top of the jet with
the critical level at the source region.
The steady-state
of all models resulted in a net decrease in the wind energy
at the topside of the jet and a net increase in wind energy
in the source region.
The ultimate energy source is then
seen to be this upper critical level.
The magnitude of the
gravity waves produced and therefore the amount of wind energy
transported depends strongly on Richardson number, the lower
the Richardson number, the greater the wave amplification
and energy transport.
However, the system remains stable
for small sources even at Richardson numbers as low as 0.200.
This would appear to be in keeping with the linear stability
calculations for shears with a ground reflector (Jones, 1968)
which state that for R. >.15 the unstable waves cannot have
frequencies less than the BrUnt frequency.
The amount of
energy transport by transient gravity wave sources can be an
order of magnitude greater than that of the input source for
R. < 0.25.
Calculations of the ground pressure fluctuations due
to these sources show that observed peak to peak surface
values of from 0.1 to 0.5 millibars can be generated by source
region rms velocities of only 0.5 to 2.0 m/s, which are well
within the limitations of the quasi-linear theory.
The quasi-linear theory has provided a framework for
studying the propagation and energy transport properties of
79
finite amplitude gravity wave transients, however,
tion of spontaneous source generation remains.
the ques-
In section IV
a theory of subharmonic wave-wave interaction was discussed.
A natural mechanism for spontaneous source generation would
seem to be the dynamic instability of highly sheared flows to
small perturbations.
A considerable volume of literature
exists on the determination of unstable regimes in terms of
Richardson number and horizontal wavenumber.
Since observed
gravity waves seemed to originate in localized regions in the
jet a small, high
shear linear segment or hyperbolic tan-
gent profile is an appropriate model for the wind near a
source.
Neutral curves, separating stable and unstable
regimes, have been found analytically for the hyperbolic tangent shear (Drazin, 1958), and numerically for the linear
shear (Jones, 1968).
A striking feature of both these curves,
however, is that all unstable modes require periods less than
a Brint period and therefore would be severely attenuated
before reaching the ground.
Observations of surface pressure
power spectra confirm this result, showing a sharp cutoff for
periods less than the BrUnt period.
Conversely, the propagat-
ing gravity waves observed would be stable in the region of
maximum shear for all Richardson numbers greater than zero.
Therefore, the source regions responsible for the observed
gravity waves do not arise spontaneously as an instability
of such waves.
Observed gravity waves with periods greater than the
Brunt period can arise from a wave-wave interaction with an
80
unstable mode through the mechanism of subharmonic generation.
For two waves having the same source region critical
level, a growing disturbance can amplify a wave with twice
its period.
The growth rate of the subharmonic is proportion-
al to the amplitude of the unstable mode which is initially
growing exponentially in accordance with linear theory.
Growth of the subharmonic extracts energy directly from the
unstable mode and will eventually stabilize it when both
waves reach approximately the same amplitude.
The calculated
growth rate of the subharmonic is shown to be very rapid compared to a wave period and appears to be almost spontaneously
generated, and since it is a propagating mode it will eventually be observed at the ground.
Calculations have also shown
that through this mechanism a background velocity noise of
1.0 cm/sec can generate surface pressure fluctuations of
approximately 1 millibar.
Lower frequency observed waves could also be generated
through wave-wave interaction as difference waves of two
unstable modes.
Although a plausible mechanism, calculations
on the three wave interaction mechanism remain an area for
further work.
A natural extension of the work presented here would
be to include the wave-wave source generation directly into
the quasi-linear theory including unstable modes and wave-wind
interaction.
One would also like to use a very realistic wind
and temperature profile and attempt direct comparison with
observed gravity waves.
Such a development would require a
81
far more sophisticated numerical simulator and large amounts
of machine time owing to the very short wavelengths and
extreme growth rates to be encountered.
APPENDIX A
NUMERICAL DETERMINATION OF THE EIGENVALUES
AND EIGENFUNCTIONS FOR STRATIFIED SHEAR FLOWS.
The linear stability for plane, stratified shear
flow is governed by the Taylor-Goldstein equation in the
Boussinesq approximation.
For vertical velocity disturbances of the form
w(x, z, t) --Q(z)ei(wt-kx) in a medium with Brunt frequency
fB and mean flow structure U(z), the governing equation is
1z
A
jR
+w
w
___
Introducing a vertical length scale d and mean
velocity scale U
06
C
then define the nondimensional parameters:
U
.... ......
U0
and scale the flow as U(z)
independent variable
U f(z/d) and introduce a new
-
=z/d.
Letting primes denote differentiation with respect to
(,
the nondimensional form of the stability equation is:
+Wr
-
C.
82
~C
The stability problem is formulated for the complex
eigenvalues c, given a, J0 and the flow structure f(z/d),
along with the boundary conditions
where (Za' Zb) are the boundary points of the system and can
be infinite.
The Hyperbolic Tangent Flow with Constant Brunt Period.
The functional form of the flow is given as
(+
0 14h(2/).
Drazin (1958) showed that all the eigenvalues will
have the same real part, that is Re(c) = 1, and that the
neutral curve for which Im(c) = o is given by
To specify the boundary value problem we then require:
a)
1IWI
R-:10 oO
()
lm
(
-
00
The stability problem is then to determine Im(c) given J
a.
and
The eigenvalues must be determined numerically away from
the neutral curve, but the existence of this curve means the
region to be examined is J' ( c 2 (1-2).
84
Transformation to Riccati Form for Eigenvalue Determination.
For numerical integration it is useful to map the
infinite domain (o, -o)
into the finite domain (1, -1).
Clearly, the transformation to use is
%J
+&VI(/d
which is monotonic and reduces f(z/d)-c to y-ilm(c).
Then
introduce the transformation
w(
[ 1-%12,
-
and the stability equation is reduced to a first-order
Riccoti equation,
zP
c
Im~c)
where
= Im(c).
where c.
3.
The initial values for $ at Y = + 1, can be determined
by requiring d$/dy to be finite there.
a)
% GM
b)
b.-N
Then
The starting values for d4/dy can be obtained from L'Hopital's
rule:
a)
b)
= I
= - do
-T-"A
+
--
-NM-W-
-
TO
For a given (a, J ) pair, the solutions for $ will
only match at y=0 if C
is an eigenvalue.
Define
85
Now, since W(zld)
+
+
0 as z
initial condition branches at y
Re (e(t) 4 0
a,
then we require that the
=
+ 1 be chosen such that
Re.( (-o))>O.
and
One can then straight forwardly integrate in from
--
+ 1 and find the zeros of
C(o
as a function of
Much work can be saved, however,
c. for given a(a, Jo) pair.
The governing equation is
with some additional analysis.
again,
cp(+tC =
and at y = +1;
:
with Re (4<:4+)) 4. 0 required.
y
= -y.
(i..._c
_--
.
Now let
-
with
Then we have
with
E()
0<_______
-
and Re (B (+1)) < 0.
Therefore, since the differential
equations and initial values are identical,
( Nw -
by uniqueness.
Therefore,
For an eigensolution,
Re
(0co))
=
4+
=
0 isrequired.
--(
,
hence
86
Further, for an eigensolution, we see that
fjRe
*~k [k
[(k-4)]
(&cU)
is
Therefore, a much easier method for finding C.
3.
available since
Re.
j(==0) =-O
so.that only one integration from either y = + 1 to
y = o is required.
Initial values of C. are then iterated
until a .zero is found in the real part of 4(y=0) which
gives the eigenvalue C. for a given (Jo, a) pair.
In principle, the nonlinear eigenuvalve problem is
now straight forward, however, if care is not taken with the
integration technique, substantial errors in the estimate of
C. can result due to the nearly singular term
-
-
Since typically C2 <<(, then
and any attempt at direct integration of the equation without
regularizing the solution is subject to significant errors in
the estimate of Ci.
Regularization of the Solution for Accurate Numerical
Integration of the Riccati form.
Let us examine the solution as
W/ >
/
as
variable -(4)
-
as
4i k(:E/d\O.
and define a corresponding Riccati
Let
87
j
-rpr
where
\A)W
and
0
)-i
+
d
satisfy
and
CL
Gi
**t-
)
exactly.
There is an analytic solution for 9
d4
; the exact solution for
where
g_%f/4-iO
is
J+
+J
A-
since
(for unstable flows
'.J- 1
Now, since the previous formulation involved 4(
so <>)so
-
$G)as
,
) define
O
will be much reduced from
near y
=0o
affording much more accurate estimations of C .
There remains to fix the constants a, b for &.
Re 0=0 at y = 0,
at y = + 1.
choose
RP (9)
=
at y = o and
Since
=
9
The eigenvalue problem is then reformulated as:
with initial conditions at y
=
+1:
h = 0
ctk=
Ir
To
__
1~__
T
88
with
(
:
::
cg
___...-_..
such that Re ($(+1)) <0.
are then found such that Re(h) = 0
The eigenvalues
at y = 0,
given values for a and J0 .
Eigenfunction Calculations
. Once an accurate determination of C
(J a) pair has been made,
for a given
direct calculation of W(y)
can be made by writing
which is governed by
P
-a d9F
+
V
2
with the initial conditions
FCq=o) ==owd
(J
23(0-4f)
c
and
APPENDIX B
BRUNT PERIOD REFLECTION CONDITIONS FOR
UNSTABLE HYPERBOLIC TANGENT FLOWS
Consider the mean flow velocity given as
then the Richardson number is
C 0S&(('/d).
CM3 =i
The condition for waves capable of propagating to
infinity is
,IQ
gfor
7-
where 52.=
all z
-C00) is the doppler-shifted frequency.
the choice of profiles, the source level is z - o, so w
For
o.
Therefore the propagation condition is
S2 > c Uc
+xVY12
'j) .
The minimum Richardson number is
Therefore the propagation condition is
To, '> kzc
-al
O/4)
and since tanh 2(z/d) < 1, then the condition is
Further, since a necessary (but not sufficient)condition for unstable modes is
89
90
all unstable modes are Brunt reflected for some finite value
of z and therefore trapped within the flow.
APPENDIX C
DISTRIBUTION OF VELOCITIES AND ENERGY DENSITIES
IN REGIONS I AND III FOR MODEL A AS A FUNCTION OF
-TIME.
CALCULATED GROUND PRESSURE FLUCTUATIONS.
The intermediate computer output for Model A is shown
in the following figures at times corresponding to t = o,
200, 400, 800, 1200, 1600, 2000, 2400, 2800, 3200, 3600 and
4000 sec.
The symbol map is
* =
100.ID
=-100.
1 =
O-.
2=
10. W, (x=0)
3=
. W/ (X=o)
L/
c
-0)
(
.
The final figure shows the calculated ground pressure
fluctuations as a function of time for the model.
91
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ACKNOWLEDGMENT
I would like to acknowledge the extensive contribution of my thesis advisor, Professor T. R. Madden, to both
the thesis and my scientific education in general.
I would
also like to thank several of my fellow graduate students for
broadening and enhancing my graduate studies, in particular,
N. Brenner, C. Moo, and A. C. Reisz, for many useful and
stimulating discussions.
I would also like to thank my wife, Nancy, for her
patience and constant support throughout my years of
graduate study.
ill
BIOGRAPHICAL NOTE
The author was born on 15 April 1946 in Los Angeles,
California, and attended Covina High School.
He received
his Bachelor's Degree in Mathematics from M.I.T. in 1967
and his Master's Degree from M.I.T. in Geophysics in 1969.
He has been employed as a Research Geophysicist at Chevron
Oil Field Research Company since September 1975.
112